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Representations and Characters of Groups, Second Edition by Martin Liebeck, Gordon James

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7

FG-homomorphisms

For groups and vector spaces, the ‘structure-preserving’ functions are, respectively, group homomorphisms and linear transformations. The analogous functions for FG-modules are called FG-homomorphisms, and we introduce these in this chapter.

FG-homomorphisms

7.1 Definition

Let V and W be FG-modules. A function ϑ: VW is said to be an FG-homomorphism if ϑ is a linear transformation and

image

In other words, if ϑ sends v to w then it sends vg to wg.

Note that if G is a finite group and ϑ: VW is an FG-homomorphism, then for all vV and , we have

since

The next result shows that FG-homomorphisms give rise to FG-submodules in ...

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